Temperature fields in an unbounded two-component cylindrical-elliptic thin plate (Q2768797)

This paper deals with the structure of stationary and non-stationary temperature fields in an unbounded two-component cylindrical-elliptic thin plate \(\Pi_{1}^{+}=\{(\xi,\eta): \xi\in(0,\xi_1)\cup(\xi_1,\infty)\); \(\eta\in[0,2\pi)\}\). In the stationary case the author obtains an explicit analytical solution of the system of Poisson equations NEWLINE\[NEWLINE{1\over\rho(\xi,\eta)}\left({\partial^2 T_{j}\over\partial\xi^2}+{\partial^2 T_{j}\over\partial\eta^2}\right)-\chi_{j}^2T_{j}(\xi,\eta)=-f_{j}(\xi,\eta),\;j=1,2NEWLINE\]NEWLINE with conditions of non-ideal thermal contact NEWLINE\[NEWLINE\left.\left[\left(b_1{\partial\over\partial\xi}+1\right)T_1(\xi,\eta)- T_2(\xi,\eta)\right]\right|_{\xi=\xi_1}=0,\;\left.\left[{\partial T_1(\xi,\eta)\over\partial\xi}- \nu_1{\partial T_2(\xi,\eta)\over\partial\xi}\right]\right|_{\xi=\xi_1}=0.NEWLINE\]NEWLINE Here \(\xi\in(\xi_{j-1},\xi_{j})\), \(\xi_0=0\), \(\xi_{2}=\infty\); \(\eta\in[0,2\pi)\), \(\chi_{j}^2\geq 0\),\( b_1\geq 0\), \(\nu_1>0\), \(\rho=2^{-1}c^2(\text{ch} 2\xi-\cos 2\eta)\). The explicit analytical solution of the non-stationary problem is also obtained.